More on Moore graphs

Let $\Delta$ be a tree such that each vertex has valency at least 3 and let $\cal A$ be a set of regular subgraphs of valency 2. In the early eighties A. Delgado and B. Stellmacher introduced the uniqueness and exchange conditions on the pair $(\Delta, {\cal A})$ and showed how they relate to generalized polygons. We modify the exchange condition and show how the modified version relates to Moore graphs. This is then used to give the isomorphism type of the amalgam of vertex stabilizers of two adjacent vertices in an $s$-arc transitive graph with trivial edge kernel and $s \geq 4$.